Beam-Columns

Learning Objectives

  • Evaluate combined axial compression and flexure using the applicable AISC interaction equation.
  • Distinguish first-order demand from second-order P−δP-\delta and P−ΔP-\Delta effects.
  • Distinguish the Direct Analysis Method from Effective Length and approximate B1/B2 second-order analysis.
  • Interpret available axial and flexural strengths as inputs produced by separate member-strength checks.
  • Recognize instability conditions instead of forcing an amplification equation beyond its valid range.
  • Coordinate member bracing strength/stiffness with the assumed unbraced lengths and structural analysis model.

Beam-Column

A member subjected to axial force and bending simultaneously. Axial compression magnifies the consequences of lateral displacement and can reduce the flexural/axial resistance available to the member.

AISC Chapter H Interaction

Available strength is already reduced strength

In the interaction equations, PcP_c, McxM_{cx}, and McyM_{cy} are available strengths, not nominal strengths. Determine them from the applicable compression and flexure provisions, including stability, LTB, local buckling, and the selected LRFD/ASD format before applying the interaction equation.

H1-type interaction for higher axial ratio

Common doubly/singly symmetric compression-plus-flexure interaction form when the axial ratio is at least 0.20.

PrPc+89(MrxMcx+MryMcy)≤1.0\frac{P_r}{P_c}+\frac{8}{9}\left(\frac{M_{rx}}{M_{cx}}+\frac{M_{ry}}{M_{cy}}\right)\le1.0

H1-type interaction for lower axial ratio

Common interaction form when the axial ratio is below 0.20.

Pr2Pc+(MrxMcx+MryMcy)≤1.0\frac{P_r}{2P_c}+\left(\frac{M_{rx}}{M_{cx}}+\frac{M_{ry}}{M_{cy}}\right)\le1.0

Confirm the Chapter H case before calculating

Chapter H contains scope conditions and additional provisions for member symmetry, torsion, unsymmetric sections, tension-plus-flexure, and other combined-force cases. Do not apply one H1 expression to every member merely because it carries axial load and moment.

Second-Order Effects

P−δP-\delta

Member-level second-order effect caused when axial compression acts through the local curvature/deflection of the member between its brace points.

P−ΔP-\Delta

System/story-level second-order effect caused when gravity/axial loads act through frame translation or story drift.

Why first-order moments are not enough

A first-order analysis evaluates equilibrium on the undeformed geometry. In a compression-loaded frame, the displaced geometry generates additional moment. As the structure approaches an instability condition, this feedback becomes increasingly important; an amplification denominator approaching zero is a warning of instability, not permission to report an arbitrarily large but finite design moment.

Three Stability-Analysis Paths Must Not Be Blended

1. Direct Analysis Method (DAM)

The Direct Analysis Method is a stability-analysis framework that directly represents second-order effects and the specified sources of geometric/material imperfection through the analysis requirements of the adopted AISC specification. It uses prescribed stiffness reductions and notional-load/imperfection treatment as applicable.

DAM is not the B1/B2 moment-amplification method. When DAM is used, the analysis itself provides the required second-order member forces for the strength checks, subject to the governing specification requirements.

2. Effective Length Method (ELM)

The Effective Length Method represents system stability partly through effective-length factors KK used in member compression strength. Its use is subject to applicability requirements and requires a defensible evaluation of frame stability/effective lengths. It should not be mixed casually with DAM stiffness/notional-load assumptions.

3. Approximate second-order analysis / moment amplification

Where permitted by the adopted specification, approximate second-order analysis separates nontranslation and translation moments and amplifies them using B1B_1 and B2B_2 factors. This is a computational route for approximating second-order demand; it is not a defining requirement of DAM.

B1/B2 Moment Amplification

Amplified required moment

Approximate separation of nontranslation and translation moment components.

Mr=B1Mnt+B2MltM_r=B_1M_{nt}+B_2M_{lt}

Member amplifier B1

Member-curvature amplification for an applicable approximate second-order case.

B1=Cm1−αPr/Pe1≥1.0B_1=\frac{C_m}{1-\alpha P_r/P_{e1}}\ge1.0

Story amplifier B2

Sway/translation amplification in the common elastic-buckling form.

B2=11−α∑Pnt/∑Pe2≥1.0B_2=\frac{1}{1-\alpha\sum P_{nt}/\sum P_{e2}}\ge1.0

Variables

SymbolDescriptionUnit
MntM_{nt}First-order moment associated with loading that does not produce lateral translation-
MltM_{lt}First-order moment associated with lateral translation-
Pe1P_{e1}Applicable member elastic buckling load for B1-
∑Pe2\sum P_{e2}Applicable story elastic buckling strength for B2-
CmC_mMoment-gradient coefficient for the applicable B1 case-
α\alphaFactor prescribed for LRFD/ASD form by the adopted provision-

Instability boundary

If an amplification denominator is zero or negative, the approximate expression does not produce a valid design demand. Treat this as an instability/out-of-scope condition and revisit the structural system, analysis, stiffness, or bracing.

Interactive second-order model

The following simulator explicitly calculates B1B_1, B2B_2, the amplified moment, and the subsequent H1 interaction. It labels B1/B2 as an approximate second-order route rather than calling it Direct Analysis.

Beam-Column Second-Order Interaction

Concept and model scope

B1/B2 moment amplification is calculated explicitly before the Chapter H interaction check

Pu/Pc0.400
B11.012
B21.250
Amplified required moment Mr276.8 kip-ft
H1 interaction0.829
Within modeled H1 limit
B1/B2 is an approximate second-order analysis path; it is not the Direct Analysis Method itself. The available strengths Pc and Mcx must come from applicable member-strength checks, and biaxial bending, torsion and other Chapter H cases require additional terms.

Bracing and Modeling Consistency

A brace needs strength and stiffness

A brace is not effective merely because a line is drawn at a node in the analysis model. Bracing must provide the strength and stiffness required to restrain the relevant member/system mode. Column flexural bracing, beam lateral bracing, beam torsional bracing, and system bracing have different behavior and requirements.

Unbraced length is directional

The effective/unbraced length for axial buckling can differ by axis, and flexural unbraced length for a beam-column can differ from the column buckling length. Model the actual restraint provided by framing and do not use one generic length for every limit state.

Design Workflow

Beam-column design sequence

  1. Establish the adopted load combinations and obtain first-/second-order required forces using a permitted stability-analysis method.
  2. Keep the selected stability method internally consistent; do not combine DAM, ELM, and B1/B2 assumptions opportunistically.
  3. Determine available axial compression strength PcP_c from the applicable Chapter E checks.
  4. Determine available flexural strengths McxM_{cx} and McyM_{cy} from the applicable Chapter F checks, including LTB/local buckling where relevant.
  5. Determine the applicable Chapter H interaction case based on member symmetry, axial-force sign/magnitude, flexure axes, and torsional effects.
  6. Evaluate the interaction using the required second-order moments/forces.
  7. Design/verify required bracing and connections so the physical structure provides the restraint assumed by the analysis and member-strength calculations.
  8. Recheck drift/serviceability, seismic system requirements, connection forces, and other project-specific limit states.
Beam-Column Stability and Interaction Check

Keep the stability-analysis method, second-order demand, available member strengths, interaction case and physical bracing aligned.

Select one permitted analysis method and follow its stiffness, imperfection, effective-length and force requirements. The Direct Analysis Method provides the required second-order forces through the analysis and is not the B1/B2 amplification method. If an approximate B1/B2 route is permitted and used, a zero or negative amplifier denominator is an instability condition; do not continue with a finite amplified design demand. Determine available axial and flexural strengths separately before applying the Chapter H case that matches the member and force state.

Beam-Column Stability and Interaction CheckKeep the stability-analysis method, second-order demand, available member strengths, interaction case and physical bracing aligned.. Select one permitted analysis method and follow its stiffness, imperfection, effective-length and force requirements. The Direct Analysis Method provides the required second-order forces through the analysis and is not the B1/B2 amplification method. If an approximate B1/B2 route is permitted and used, a zero or negative amplifier denominator is an instability condition; do not continue with a finite amplified design demand. Determine available axial and flexural strengths separately before applying the Chapter H case that matches the member and force state.. Design a beam-column → Set load combinations, geometry, restraint and code basis; Set load combinations, geometry, restraint and code basis → Select one permitted stability-analysis method; Select one permitted stability-analysis method → Is an approximate B1/B2 method being used?; Is an approximate B1/B2 method being used? — Yes → Are all applicable amplifier denominators positive?; Is an approximate B1/B2 method being used? — No → Use the selected method's required forces without B1/B2; Use the selected method's required forces without B1/B2 → Obtain required forces consistent with the selected method; Are all applicable amplifier denominators positive? — Yes → Obtain required forces consistent with the selected method; Are all applicable amplifier denominators positive? — No → Treat as unstable; review system, stiffness or bracing; Treat as unstable; review system, stiffness or bracing → Review stability assumptions before proceeding; Obtain required forces consistent with the selected method → Determine available Chapter E and F strengths; Determine available Chapter E and F strengths → Select the applicable Chapter H interaction case; Select the applicable Chapter H interaction case → Does the governing interaction meet its limit?; Does the governing interaction meet its limit? — Yes → Verify bracing, connections and project-specific checks; Does the governing interaction meet its limit? — No → Revise the member; repeat affected checks; Verify bracing, connections and project-specific checks → Do modeled restraints match the physical design?; Do modeled restraints match the physical design? — Yes → Record analysis method, strengths and interaction result; Do modeled restraints match the physical design? — No → Revise the model or system; repeat affected checks; Record analysis method, strengths and interaction result → Complete the beam-column design

Design a beam-column → Set load combinations, geometry, restraint and code basis; Set load combinations, geometry, restraint and code basis → Select one permitted stability-analysis method; Select one permitted stability-analysis method → Is an approximate B1/B2 method being used?; Is an approximate B1/B2 method being used? — Yes → Are all applicable amplifier denominators positive?; Is an approximate B1/B2 method being used? — No → Use the selected method's required forces without B1/B2; Use the selected method's required forces without B1/B2 → Obtain required forces consistent with the selected method; Are all applicable amplifier denominators positive? — Yes → Obtain required forces consistent with the selected method; Are all applicable amplifier denominators positive? — No → Treat as unstable; review system, stiffness or bracing; Treat as unstable; review system, stiffness or bracing → Review stability assumptions before proceeding; Obtain required forces consistent with the selected method → Determine available Chapter E and F strengths; Determine available Chapter E and F strengths → Select the applicable Chapter H interaction case; Select the applicable Chapter H interaction case → Does the governing interaction meet its limit?; Does the governing interaction meet its limit? — Yes → Verify bracing, connections and project-specific checks; Does the governing interaction meet its limit? — No → Revise the member; repeat affected checks; Verify bracing, connections and project-specific checks → Do modeled restraints match the physical design?; Do modeled restraints match the physical design? — Yes → Record analysis method, strengths and interaction result; Do modeled restraints match the physical design? — No → Revise the model or system; repeat affected checks; Record analysis method, strengths and interaction result → Complete the beam-column design

  • Design a beam-column: terminator
  • Set load combinations, geometry, restraint and code basis: input output
  • Select one permitted stability-analysis method: process
  • Is an approximate B1/B2 method being used?: decision
  • Use the selected method's required forces without B1/B2: process
  • Are all applicable amplifier denominators positive?: decision
  • Obtain required forces consistent with the selected method: process
  • Treat as unstable; review system, stiffness or bracing: process
  • Determine available Chapter E and F strengths: process
  • Select the applicable Chapter H interaction case: process
  • Does the governing interaction meet its limit?: decision
  • Revise the member; repeat affected checks: terminator
  • Verify bracing, connections and project-specific checks: process
  • Do modeled restraints match the physical design?: decision
  • Record analysis method, strengths and interaction result: document
  • Revise the model or system; repeat affected checks: terminator
  • Review stability assumptions before proceeding: terminator
  • Complete the beam-column design: terminator
Key Takeaways
  • Beam-column design couples member strength with structural stability analysis.
  • P−δP-\delta is member curvature; P−ΔP-\Delta is system/story translation.
  • Direct Analysis, Effective Length, and B1/B2 approximate second-order analysis are distinct methods and must not be blended casually.
  • The interaction equation uses available member strengths and required second-order demands.
  • An amplification denominator at or below zero is an instability condition, not a finite design result.
  • Bracing strength/stiffness and actual restraint geometry must match the analysis model.